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Similar Polygons

Similar Polygons. Section 7-2. Objective. Identify similar polygons. Key Vocabulary. Similar polygons Similarity ratio Scale factor. Theorems. 7.1 Perimeters of Similar Polygons. Similar Triangles. What is Similarity?. Not Similar. Similar. Similar. Not Similar. NOT Similar.

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Similar Polygons

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  1. Similar Polygons Section 7-2

  2. Objective • Identify similar polygons.

  3. Key Vocabulary • Similar polygons • Similarity ratio • Scale factor

  4. Theorems • 7.1 Perimeters of Similar Polygons

  5. Similar Triangles What is Similarity? Not Similar Similar Similar Not Similar

  6. NOT Similar Similarity Figures that have the same shape but not necessarily the same size are similar figures. But what does “same shape mean”? Are the two heads similar?

  7. Similarity Similar shapes can be thought of as enlargements or reductions with no irregular distortions. • So two shapes are similar if one can be enlarged or reduced so that it is congruent to the original.

  8. Similar Polygons • When polygons have the same shape but may be different in size, they are called similar polygons. • We express similarity using the symbol, ~. (i.e. ΔABC ~ ΔPRS)

  9. Example - Similar Polygons Figures that are similar(~) have the same shape but not necessarily the same size.

  10. Similar Polygons • The order of the vertices in a similarity statement is very important. It identifies the corresponding angles and sides of the polygons.ΔABC ~ ΔPRS A P, B R, C S AB = BC = CA PR RS SP

  11. Similar Polygons Two polygons are similar polygonsiff the corresponding angles are congruent and the corresponding sides are proportional. Similarity Statement: Corresponding Angles: Statement of Proportionality:

  12. IMPORTANT Writing a similarity statement is like writing a congruence statement—be sure to list corresponding vertices in the same order.

  13. Similarity and Congruence • If two polygons are congruent, they are also similar. • All of the corresponding angles are congruent, and the lengths of the corresponding sides have a ratio of 1:1.

  14. Example 1 • If ΔABC ~ ΔRST, list all pairs of congruent angles and write a proportion that relates the corresponding sides.

  15. Example 1 • Use the similarity statement. ΔABC~ ΔRST Answer: Congruent Angles: A R,B S,C T

  16. A.HK ~ QR B. C.K ~ R D.GHK ~ QPR Your Turn: • If ΔGHK ~ ΔPQR, determine which of the following similarity statements is not true.

  17. Example 2 . . PRQ ~ STU. a. List all pairs of congruent angles. b. Write the ratios of the corresponding sides in a statement of proportionality. 16 12 8 ST TU US ST US TU 4 4 4 QP PR 10 RQ 20 15 PR RQ QP 5 5 5 c. Check that the ratios of corresponding sides are equal. SOLUTION 4 The ratios of corresponding sides are all equal to 5 a. PS,RT,andQU. b. = = = = = = = = c. , , and

  18. Example 3 SOLUTION Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A. 1. Check whether the corresponding angles are congruent. From the diagram, you can see that G M,H  K, and J  L. Therefore, the corresponding angles are congruent.

  19. Example 3 All three ratios are equal, so the corresponding side lengths are proportional. By definition, the triangles are similar. GHJ ~ MKL. ANSWER 4 . The scale factor of Figure B to Figure A is 3 MK 12 12 ÷ 3 4 = = = GH 9 9 ÷ 3 3 KL 16 16 ÷ 4 4 = = = HJ 12 ÷ 4 3 12 LM 20 20 ÷ 5 4 = = = JG 15 ÷ 5 3 15 2. Check whether the corresponding side lengths are proportional.

  20. Your Turn: 3 ANSWER yes;XYZ ~ DEF; 2 no ANSWER Determine whether the polygons are similar. If they are similar, write a similarity statement and find the scale factor of Figure B to Figure A. 1. 2. 12 9 ≠ 10 6

  21. Example 4a • A. MENUSTan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu:

  22. Example 4a Step 1Compare corresponding angles. Since all angles of a rectangle are right angles and right angles are congruent, corresponding angles are congruent. Step 2 Compare corresponding sides. Original New

  23. Example 4a Answer:Since corresponding sides are not proportional, ABCD is not similar to FGHK. So, the menus are not similar.

  24. Example 4b • B. MENUSTan is designing a new menu for the restaurant where he works. Determine whether the size for the new menu is similar to the original menu. If so, write the similarity statement and scale factor. Explain your reasoning. Original Menu: New Menu:

  25. Example 4b Original Step 1Compare corresponding angles. Since all angles of a rectangle are right angles and right angles are congruent, corresponding angles are congruent. Step 2 Compare corresponding sides. New

  26. Answer:Since corresponding sides are proportional, ABCD ~ RSTU. So the menus are similar with a scale factor of . 4 __ 5 Example 4b

  27. Your Turn: A.Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. Original: New: A.BCDE ~ FGHI, scale factor = B.BCDE ~ FGHI, scale factor = C.BCDE ~ FGHI, scale factor = D.BCDE is not similar to FGHI.

  28. A.BCDE ~ WXYZ, scale factor = B.BCDE ~ WXYZ, scale factor = C.BCDE ~ WXYZ, scale factor = D.BCDE is not similar to WXYZ. 1 4 3 __ __ __ 2 5 8 Your Turn: B.Thalia is a wedding planner who is making invitations. Determine whether the size for the new invitations is similar to the original invitations used. If so, choose the correct similarity statement and scale factor. Original: New:

  29. Example 5a A.The two polygons are similar. Find x. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV

  30. 9 __ 2 Answer: x = Example 5a Write proportions to find x. Similarity proportion Cross Products Property Multiply. Divide each side by 4.

  31. Example 5b B.The two polygons are similar. Find y. Use the congruent angles to write the corresponding vertices in order. polygon ABCDE ~ polygon RSTUV

  32. 13 __ 3 Answer: y = Example 5b Similarity proportion AB = 6, RS = 4, DE = 8, UV = y + 1 Cross Products Property Multiply. Subtract 6 from each side. Divide each side by 6 and simplify.

  33. Your Turn: A. The two polygons are similar. Solve for a. A.a = 1.4 B.a = 3.75 C.a = 2.4 D.a = 2

  34. Your Turn: B. The two polygons are similar. Solve for b. A. 1.2 B. 2.1 C. 7.2 D. 9.3

  35. Identifying Similar Triangles • When only two congruent angles of a triangle are given, remember that you can use the Third Angles Theorem to establish that the remaining corresponding angles are also congruent. • Example:

  36. T Example 6 Determine whether the pair of figures is similar.Justify your answer. Thus, all the corresponding angles are congruent.

  37. T Answer: The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, so Example 6 Now determine whether corresponding sides are proportional. The ratios of the measures of the corresponding sides are equal.

  38. Determine whether the pair of figures is similar.Justify your answer.a. Your Turn:

  39. Answer: Both triangles are isosceles with base angles measuring 76º and vertex angles measuring 28º. The ratio of the measures of the corresponding sides are equal and the corresponding angles are congruent, Your Turn:

  40. Determine whether the pair of figures is similar.Justify your answer. b. Your Turn: Answer: Only one pair of angles are congruent, so the triangles are not similar.

  41. Scale Factor • In similar polygons, the ratio of two corresponding sides is called a scale factor. • The scale factor depends on the order of comparison. • What is the scale factor of the similar polygons shown?

  42. Scale Factor • The scale factor between two similar polygons is sometimes called the similarity ratio. • Scale factors are usually given for models of real-life objects.

  43. Example 7 An architect prepared a 12-inch model of a skyscraper to look like a real 1100-foot building. What is the scale factor of the model compared to the real building? Before finding the scale factor you must make sure that both measurements use the same unit of measure. 1100(12) = 13,200 inches Scale factor

  44. Answer:The ratio comparing the two heights is The scale factor is , which means that the model is the height of the real skyscraper. Example 7

  45. Answer: Your Turn: A space shuttle is about 122 feet in length. The Science Club plans to make a model of the space shuttle with a length of 24 inches. What is the scale factor of the model compared to the real space shuttle?

  46. Example 8 The two polygons are similar. Find the scale factor of polygon ABCDE to polygon RSTUV.

  47. Answer: Example 8 The scale factor is the ratio of the lengths of any two corresponding sides.

  48. Answer: a. Write a similarity statement. Then find a, b, and ZO. b.Find the scale factor of polygon TRAP to polygon . Answer:; Your Turn: Your Turn: The two polygons are similar.

  49. Write proportions for finding side measures. Let one long side of each WXYZ and PQRS be and one short side of each WXYZ and PQRS be Example 9 Rectangle WXYZ is similar to rectangle PQRSwith a scale factor of 1.5. If the length and width of rectangle PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ?

  50. Answer: Example 9

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